Multibody System Dynamics

, Volume 40, Issue 4, pp 375–405 | Cite as

Simulation of a helicopter’s main gearbox semiactive suspension with bond graphs

  • Benjamin BoudonEmail author
  • François Malburet
  • Jean-Claude Carmona


This paper presents a bond graph model of a helicopter’s semiactive suspension and the associated simulations. The structural and modular approach proposed with bond graph permits a systematic modeling of mechatronic multibody systems. This approach was carried out thanks to the use of the singular perturbation method, which is a variant of penalty formulation. The model is then built as an assembly of components or modules (rigid bodies and compliant kinematic joints) by following the structure of the actual system.

The bond graph model of the passive suspension with fixed flapping masses has been verified with another multibody tool for three different excitations (pumping, roll, and yaw). Next, the passive model, augmented with electrical actuators and controllers, is called the semiactive suspension model. Simulations on the semiactive suspension model have been conducted.


Multibody systems (MBS) Closed kinematic chain (CKC) Bond graph (BG) Helicopter Mechanical vibrations 20-sim 



This research work received support from the Chair “Dynamics of complex mechanical systems—EADS Corporate Foundation—Arts et Métiers Paris Tech and Ecole Centrale de Marseille.” Thanks to Paul B.T. Weustink working at Controllab Products for his help on the use of complementary tools of 20-sim software.


  1. 1.
    Schiehlen, W.: Research trends in multibody system dynamics. Multibody Syst. Dyn. 18, 3–13 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    van Amerongen, J., Breedveld, P.C.: Modelling of physical systems for the design and control of mechatronic systems. Annu. Rev. Control 27(1), 87–117 (2003) CrossRefGoogle Scholar
  3. 3.
    Xu, W., Liu, Y., Liang, B., Wang, X., Xu, Y.: Unified multi-domain modelling and simulation of space robot for capturing a moving target. Multibody Syst. Dyn. 23, 293–331 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Paynter, H.M.: Analysis and Design of Engineering Systems. MIT Press, Cambridge (1961) Google Scholar
  5. 5.
    Borutsky, W.: Bond Graph Methodology—Development and Analysis of Multidisciplinary Dynamic System Models. Springer, London (2010) Google Scholar
  6. 6.
    Sass, L., McPhee, J., Schmitke, C., Fisette, P., Grenier, D.: A comparison of different methods for modelling electromechanical multibody systems. Multibody Syst. Dyn. 12, 209–250 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Tiernego, M.J.L., Bos, A.M.: Modelling the dynamics and kinematics of mechanical systems with multibond graphs. J. Franklin Inst. 319, 37–50 (1985) CrossRefGoogle Scholar
  8. 8.
    Bos, A.M.: Multibody systems in terms of multibond graphs with application to a motorcycle multibody system. Ph.D. Thesis, University of Twente, Enschede, The Netherlands (1986) Google Scholar
  9. 9.
    Abadie, V., Guillemard, F., Rault, A.: Apport du bond graph dans la démarche mécatronique appliquée à l’automobile. In: Dauphin-Tanguy, G. (ed.) Les Bond Graphs. Hermès Sciences, Paris (2000) Google Scholar
  10. 10.
    Zeid, A., Chung, C.-H.: Bond graph modeling of multibody systems: a library of three-dimensional joints. J. Franklin Inst. 329, 605–636 (1992) CrossRefzbMATHGoogle Scholar
  11. 11.
    Zeid, A.A., Overholt, J.L.: Singularly perturbed formulation: explicit modeling of multibody systems. J. Franklin Inst. 332(1), 21–45 (1995). doi: 10.1016/0016-0032(95)00002-F CrossRefzbMATHGoogle Scholar
  12. 12.
    Wang, J., Gosselin, C., Cheng, L.: Modeling and simulation of robotic systems with closed kinematic chains using the virtual spring approach. Multibody Syst. Dyn. 7, 145–170 (2001) CrossRefzbMATHGoogle Scholar
  13. 13.
    Krysinski, T., Malburet, F.: Mechanical Vibrations: Active and Passive Control. Wiley, New York (2006) zbMATHGoogle Scholar
  14. 14.
    Henderson, J.-P.: Vibration isolation for rotorcraft using electrical actuation. Ph.D. Thesis, Mechanical Engineering, University of Bath (2012) Google Scholar
  15. 15.
    Karnopp, D.C., Margolis, D.L., Rosemberg, R.C.: System Dynamics—Modeling and Simulation of Mechatronics Systems. Wiley, New York (2000) Google Scholar
  16. 16.
    Tiller, M.: Introduction to Physical Modeling with Modelica. Kluwer Academic, Dordrecht (2001) CrossRefGoogle Scholar
  17. 17.
    Marquis-Favre, W., Jardin, A.: Bond graphs and inverse modeling for mechatronic system design. In: Borutsky, W. (ed.) Bond Graph Modeling of Engineering Systems. Springer, Berlin (2011) Google Scholar
  18. 18.
    Felez, J., Romero, G., Maroto, J., Martinez, M.L.: Simulation of multi-body systems using multi-bond graph. In: Borutsky, W. (ed.) Bond Graph Modeling of Engineering Systems. Springer, Berlin (2011) Google Scholar
  19. 19.
    Karnopp, D.: Power-conserving transformations: physical interpretations and applications using bond graphs. J. Franklin Inst. 288, 175–201 (1969) CrossRefGoogle Scholar
  20. 20.
    Rosenberg, R.C.: Multiport models in mechanics. J. Dyn. Syst. Meas. Control 94, 206–212 (1972) CrossRefGoogle Scholar
  21. 21.
    Bonderson, L.S.: Vector bond graphs applied to one-dimensional distributed systems. J. Dyn. Syst. Meas. Control 1, 75–82 (1975) CrossRefGoogle Scholar
  22. 22.
    Breedveld, P.C.: Multibond graph elements in physical systems theory. J. Franklin Inst. 319, 1–36 (1985) CrossRefGoogle Scholar
  23. 23.
    Felez, J., Vera, C., San Jose, I., Cacho, R.: BONDYN: a bond graph based simulation program for multibody systems. J. Dyn. Syst. Meas. Control 112, 717–727 (1990) CrossRefGoogle Scholar
  24. 24.
    Van Dijk, J., Breedveld, P.C.: Simulation of system models containing zero-order causal paths—I. Classification of zero-order causal paths. J. Franklin Inst. 328, 959–979 (1991) CrossRefzbMATHGoogle Scholar
  25. 25.
    Van Dijk, J., Breedveld, P.C.: Simulation of system models containing zero-order causal paths—II. Numerical implications of class 1 zero-order causal paths. J. Franklin Inst. 328, 981–1004 (1991) CrossRefzbMATHGoogle Scholar
  26. 26.
    Marquis-Favre, W., Scavarda, S.: Bond graph representation of multibody systems with kinematic loops. J. Franklin Inst. 335B, 643–660 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Samin, J.C., Bruls, O., Collard, J.F., Sass, L., Fisette, P.: Multiphysics modeling and optimization of mechatronic multibody systems. Multibody Syst. Dyn. 18, 345–373 (2007) CrossRefzbMATHGoogle Scholar
  28. 28.
    Ersal, T., Stein, J.L., Louca, L.S.: A bond graph based modular modeling approach towards an automated modeling environment for reconfigurable machine tools. In: IMAACA, Genoa, Italy (2004) Google Scholar
  29. 29.
    Behzadipour, S., Khajepour, A.: Causality in vector bond graphs and its application to modeling of multi-body dynamic systems. Simul. Model. Pract. Theory 14, 279–295 (2005) CrossRefzbMATHGoogle Scholar
  30. 30.
    Marquis-Favre, W.: Simulation des mécanismes: Résolution des équations dans les logiciels. Techniques de l’ingénieur (2007) Google Scholar
  31. 31.
    Uchida, T.K.: Real-time dynamic simulation of constrained multibody systems using symbolic computation. Ph.D. Thesis, University of Waterloo, Ontario (2011) Google Scholar
  32. 32.
    Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical solution of initial-value problems in differential-algebraic equations, vol. 14. SIAM, Philadelphia (1996) zbMATHGoogle Scholar
  33. 33.
    Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1, 1–16 (1972) MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Garcia, J., Jalon, D.E., Bayo, E.: Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge. Springer, New York (1994) CrossRefGoogle Scholar
  35. 35.
    Rideout, G.: System partitioning and physical domain proper modeling through assessment of power-conserving model structure. Ph.D. Thesis, University of Michigan (2004) Google Scholar
  36. 36.
    Rahman, T., Rideout, G., Krouglicof, N.: Evaluation of dynamic performance of non-spherical parallel orientation manipulators through bond graph multi-body simulation. In: ICBGM, Genoa, Italy (2012) Google Scholar
  37. 37.
    Lalanne, C.: Mechanical Vibration and Shock Analysis. Wiley, New York (2014) Google Scholar
  38. 38.
    Marquis-Favre, W.: Contribution à la représentation bond graph des systèmes mécaniques multicorps. Ph.D. Thesis, INSA de Lyon (1997) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Benjamin Boudon
    • 1
    Email author
  • François Malburet
    • 2
  • Jean-Claude Carmona
    • 2
  1. 1.ISM UMR 7287Aix-Marseille Université, CNRSMarseille Cedex 09France
  2. 2.Arts-et-Métiers ParisTechCNRS, LSISAix-en-ProvenceFrance

Personalised recommendations